Reply: This is verso good objection. However, the difference between first-order and tsdating higher-order relations is relevant here. Traditionally, similarity relations such as x and y are the same color have been represented, durante the way indicated in the objection, as higher-order relations involving identities between higher order objects (properties). Yet this treatment may not be inevitable. Mediante Deutsch (1997), an attempt is made sicuro treat similarity relations of the form ‘\(x\) and \(y\) are the same \(F\)’ (where \(F\) is adjectival) as primitive, first-order, purely logical relations (see also Williamson 1988). If successful, verso first-order treatment of similarity would esibizione that the impression that identity is prior to equivalence is merely a misimpression – paio preciso the assumption that the usual higher-order account of similarity relations is the only option.

Objection 6: If on day 3, \(c’ = s_2\), as the text asserts, then by NI, the same is true on day 2. But the text also asserts that on day 2, \(c = s_2\); yet \(c \ne c’\). This is incoherent.

Objection 7: The notion of incomplete identity is incoherent: “If verso cat and one of its proper parts are one and the same cat, what is the mass of that one cat?” (Burke 1994)

Reply: Young Oscar and Old Oscar are the same dog, but it makes in nessun caso sense to ask: “What is the mass of that one dog.” Given the possibility of change, identical objects may differ con mass. On the divisee identity account, that means that distinct logical objects that are the same \(F\) may differ con mass – and may differ with respect to verso host of other properties as well. Oscar and Oscar-minus are distinct physical objects, and therefore distinct logical objects. Distinct physical objects may differ in mass.

Objection 8: We can solve the paradox of 101 Dalmatians by appeal preciso verso notion of “almost identity” (Lewis 1993). We can admit, con light of the “problem of the many” (Unger 1980), that the 101 dog parts are dogs, but we can also affirm that the 101 dogs are not many; for they are “almost one.” Almost-identity is not a relation of indiscernibility, since it is not transitive, and so it differs from imparfaite identity. It is verso matter of negligible difference. Verso series of negligible differences can add up puro one that is not negligible.

Let \(E\) be an equivalence relation defined on per arnesi \(A\). For \(x\) sopra \(A\), \([x]\) is the attrezzi of all \(y\) in \(A\) such that \(E(incognita, y)\); this is the equivalence class of quantitativo determined by Ancora. The equivalence relation \(E\) divides the serie \(A\) into mutually exclusive equivalence classes whose union is \(A\). The family of such equivalence classes is called ‘the partition of \(A\) induced by \(E\)’.

## 3. Incomplete Identity

Garantis that \(L’\) is some fragment of \(L\) containing verso subset of the predicate symbols of \(L\) and the identity symbol. Let \(M\) be verso structure for \(L’\) and suppose that some identity statement \(verso = b\) (where \(a\) and \(b\) are individual constants) is true mediante \(M\), and that Ref and LL are true durante \(M\). Now expand \(M\) preciso a structure \(M’\) for per richer language – perhaps \(L\) itself. That is, garantit we add some predicates puro \(L’\) and interpret them as usual durante \(M\) esatto obtain an expansion \(M’\) of \(M\). Garantis that Ref and LL are true per \(M’\) and that the interpretation of the terms \(a\) and \(b\) remains the same. Is \(verso = b\) true mediante \(M’\)? That depends. If the identity symbol is treated as a logical constant, the answer is “yes.” But if it is treated as per non-logical symbol, then it can happen that \(per = b\) is false in \(M’\). The indiscernibility relation defined by the identity symbol in \(M\) may differ from the one it defines con \(M’\); and sopra particular, the latter may be more “fine-grained” than the former. Con this sense, if identity is treated as a logical constant, identity is not “language incomplete;” whereas if identity is treated as per non-logical notion, it \(is\) language divisee. For this reason we can say that, treated as a logical constant, identity is ‘unrestricted’. For example, let \(L’\) be a fragment of \(L\) containing only the identity symbol and a solo one-place predicate symbol; and suppose that the identity symbol is treated as non-logical. The espressione

## 4.6 Church’s Paradox

That is hard puro say. Geach sets up two strawman candidates for absolute identity, one at the beginning of his colloque and one at the end, and he easily disposes of both. Per between he develops an interesting and influential argument puro the effect that identity, even as formalized durante the system FOL\(^=\), is divisee identity. However, Geach takes himself sicuro have shown, by this argument, that absolute identity does not exist. At the end of his initial presentation of the argument con his 1967 paper, Geach remarks: